Data Analysis and Statistical Methods Statistics 651

Transcription

1 Data Analysis and Statistical Methods Statistics Lecture 10 (MWF) Checking for normality of the data using the QQplot Suhasini Subba Rao

2 Review of previous lecture We calculated probabilities of a normal distribution by standardisation. Example Suppose X N( 3, 0.5), what is P (X 3.5)? Standardise: P (X 3.5) = P ( X ) = P (Z 0.707) (by using the normal tables). We note that when we calculate we are going from a nonstandard normal X N( 3, 0.5) to a standard normal, hence Z = , where Z N(0, 1). We also did the reverse of this finding the values on the x-axis where P (X x) = 0.8, when X N(6, 7) (for example). In this case we had to standardise: P ( X 6 7 x 9 7 ) = 0.8. Look up in the tables the z-value that corresponds to 0.8. This is Therefore x 9 7 = 0.85 and solve for x. 1

3 Checking for Normality (a very rough check) Suppose x 1,..., x n is a sample from a normal distribution with mean µ and variance σ 2. First we order them from the smallest number to the largest number: x (1),..., x (n). Estimate the mean and standard deviations from the data; x and s. Plot all the observations on a number line. Locate the mean x on this line and also the intervals: [ x s, x + s], [ x 2s, x + 2s] and [ x 3s, x + 3s]. If the observations came from a normal, then Roughly 68% of the observations should lie in the interval [ x s, x+s]. 2

4 95% of the observations should lie in the interval [ x 2s, x + 2s]. 99.7% of the observations should lie in the interval [ x 3s, x + 3s]. Remember this means counting the number of points in each interval, and dividing it by the total number of observations. This is an extremely rough way to check for normality. There can exist weird non-normal distributions where the following: Roughly 68% of the observations should lie in the interval [ x s, x+s]. 95% of the observations should lie in the interval [ x 2s, x + 2s]. 99.7% of the observations should lie in the interval [ x 3s, x + 3s]. could be true! 3

5 Motivating the QQplot Lecture 10 (MWF) QQplots We need to find a more accurate method (which is close in idea to the counting in an interval). This motivates the idea of the QQplot. Roughly speaking the QQplots orders the data from the smallest to the largest and plots the data against corresponding normal quantile. Data X 1,..., X n ordered from smallest to largest X (1),..., X (n). Plot X (i) against the i/n quantile of the normal distribution (omitting the first and last observations). If the data comes from a normal distribution (with the mean and variance estimated from the data) the data (empirical quantiles) will match the normal quantiles, and plot should lie on a straightline (on the x = y line). This is the QQplot. 4

6 Checking for normality: The QQ plot This plots what has been described above. The QQplot consists of points and a straight 45 degree line. X X X X X (5) (4) (3) (2) (1)..... x=y line y y y y y (1) (2) (3) (4) (5) If the points tend to lie on the straightline, then this suggests the observations come from a normal distribution. 5

7 Example: Antarctic maximum temperature QQplot It would appear that the maximum temperatures are close to normal. The mean of this data is about 4.5 and the standard deviation is

8 Using this information we can calculate the probabilities. This months maximum temperature is 7 degrees, what is its percentile? Answer P (X 7) = P (Z (7 4.5)/2.16) = This tells me the temperature is in the 87% percentile (using the normal approximation). Based on the data the proportion of temperatures less than 7 degrees is about 86.5% which fits the calculation made using normal approximation of the data very well. 7

9 Example: Antarctic minimum temperature QQplot The minimum temperatures appear to be far from normal. We know that the mean and standard deviation of this data is 13.8 degrees and 9.3 respectively. 8

10 If we use normality of the data to calculate the chance of the temperature being less than -10 we have P (X 10) = P (Z = ) = (about 65.4%). Based on the data the proportion of temperatures less than -10 degrees is about 55%, which is quite different to the proportion calculated using the normal approximation. 9

11 Interpretating a QQ-plot Lecture 10 (MWF) QQplots Some experienced statisticans have shaman like powers when it comes to interpretating QQ-plots. You don t need them, but it is good to have a feel of them. There are two main features you need to look for; Left Skew. This means the distribution is not symmetric. Find the mode (the heightest point of the distribution). The right of the mode should be shorter than the left of the mode. Right Skew. This means the distribution is not symmetric. Find the mode (the heightest point of the distribution). The right of the mode should be longer than the left of the mode. Heavy tails. This means that the probability of large numbers if much more likely than a normal distribution. For example for a 10

12 normal distribution most the observations 98% lie within the interval [ x 3s, x + 3s]. For a heavy tail distribution a far smaller proportion lie in this interval. 11

13 Skewed distributions Lecture 10 (MWF) QQplots A right skewed distribution (red) has a long right tail (green is normal). For a left skewed distribution the QQ-plot is the mirror image along the 45 degree line (arch going upwards and towards the left). 12

14 A right skewed distribution and it s QQplot This is right skewed and we see the qqplots looks like a U. 13

15 QQ-plot of a left skewed distribution The above is indicates a left skewed distribution. The points are arched, going from the below the 45 degree line across it and down again. 14

16 Heavy tail distribution Lecture 10 (MWF) QQplots Has much thicker tails than a normal distribution (the blue are the tails of a normal and red are the tails of a thick tail). 15

17 QQ-plot of a heavy tailed distribution The plot is like an S. On the left of the plot it is left of the 45 degree line and then towards the right it goes to being right of the 45 degree line. 16

18 What does thick tailed distribution mean?? Look at the histogram of the following data set (size 200 observations). Look at the proportion of points outside one/two and three standard deviations of the mean (compare with 68%, 95% and 99.8%). It is a lot more than the normal distribution. Look at the tails, it is higher (thicker) than the normal distribution. 17

19 The corresponding QQplot Below we make a QQplot of the above data set. Lecture 10 (MWF) QQplots The S shape suggests the distribution has thick tails. 18

20 QQplot of the original M&M data It is clearly non-normal. First the horizontal lines that we see is because the data is integer valued (not normal), second it has a strange, shape that does not look at all like it lies on the x=y line. 19

21 QQplot of the average of 5 M&M bags Does not look normal, but certainly the qqplot looks closer x=y then the previous plot. 20

22 QQplot of the average of 10 M&M bags Again it does not look normal, but it does look more normal than the original data. Remember we only have 17 averages, which may explain why the histogram of the averages looks more flat than bell-shaped. 21

23 QQplot of binary data Let us return to the example of people liking apple juice. 100 people were interviewed and each person was asked whether they like apple juice or not (1=yes, 0 = no). Here is the data 22

24 34% of this sample liked apple juice. This data is binary (not normal!), this is why you see the two lines. It is clearly not normal, and you cannot make it more normal by increasing the sample size. What does become normal is the sample proportion (which in this case is 34%) - this is due to the CLT, which we discuss in lecture 12. But only when the sample size is relatively large. 23

25 Transforming Data Lecture 10 (MWF) QQplots If the data is far from normal we often do a transformation of it to make it have less outliers and less skewed. Standard transforms are (for positive data); The log transform; X i log X i = Y i. The variance of the transformed observation tends to be less than the variance of the original observation (sometimes this transformation is called variance stablisation ). Often used when the sample mean and sample variance of X are similar. The power transform; X i X β i = Y i (where 0 < β < 1). This transformation tends to control outliers and unskews the data. 24

26 Left is a QQplot of the original data and the right is the QQplot of the square root of the data (ie. X i X i = X 1/2 i ). Observe how the square root of the data is still skewed - but it is less skewed than the original data. Reducing skewness in data is very useful way of making the CLT work for smaller sample sizes (see later). 25

27 QQ plots and testing for normality There are statistical tests (I have not defined this yet) for checking normality. One of the most famous ones is called the Kolmogorov- Smirnov test. QQ plots for other distributions It is possible by make a QQplot for other distributions. That is to check whether the observations are drawn from another distribution of interest. The QQplot must be modified to the new distribution (where the quantiles of distribution are compared with the ordered data). If you want to know how please ask me. Again the Kolmogorov-Smirnov test can be used to check whether the observations come from the distribution of interest. 26